You should use free competitor website research tool. Some good tools are:
Open Site Explorer, Google AdWords Keyword Tool, SEMRush, Alexa.

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Well, I don't why you are having trouble in knowing things as I can see that you are a webmaster of some site but still you seemed to be running onto trouble which I actually don't understand, but anyway let me clear this for you to know about your competitor what you can do is open up the search engine let us just say google.com and type the keyword which you are trying to rank yourself upon !! And after entering that keyword try searching yourself over that Keyword, once you locate yourself now start looking above the search results, all the websites appearing above you are the competitors.

Website Competitive Analysis - 8 Tools to Research Your Competition you can check this site out.

Use Ahref. They are the best,believe me.

Hi, By searching in your keywords in Search engine for that keyword which site is ranking on top those are your competitors...